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On the minimum exit rate for a diffusion process pertaining to a chain of distributed control systems with random perturbations
In this paper, we consider the problem of minimizing the exit rate with which
a diffusion process pertaining to a chain of distributed control systems, with
random perturbations, exits from a given bounded open domain. In particular, we
consider a chain of distributed control systems that are formed by
subsystems (with ), where the random perturbation enters only in the
first subsystem and is then subsequently transmitted to the other subsystems.
Furthermore, we assume that, for any , the
distributed control systems, which is formed by the first subsystems,
satisfies an appropriate H\"ormander condition. As a result of this, the
diffusion process is degenerate, in the sense that the infinitesimal generator
associated with it is a degenerate parabolic equation. Our interest is to
establish a connection between the minimum exit rate with which the diffusion
process exits from the given domain and the principal eigenvalue for the
infinitesimal generator with zero boundary conditions. Such a connection allows
us to derive a family of Hamilton-Jacobi-Bellman equations for which we provide
a verification theorem that shows the validity of the corresponding optimal
control problems. Finally, we provide an estimate on the attainable exit
probability of the diffusion process with respect to a set of admissible
(optimal) Markov controls for the optimal control problems.Comment: 12 Pages. (Additional Note: This work is, in some sense, a
continuation of our previous paper arXiv:1408.6260.
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